Proving a subspace

  • #1

Homework Statement



Prove that the set of all n x n matrices A such that AB = BA for a fixed n x n matrix B, is a subspace of Mnn.

Homework Equations



u + v is in the same vector space as u and v.
ku is in the same vector space as u, where k is any real number.

The Attempt at a Solution



I am drawn to think of a diagonal matrix when I think of this question. And if I multiply a diagonal matrix by a scalar, it can only be a diagonal matrix or the zero matrix, either way, it AB still equals BA. Similarly, adding two diagonal matrices obtains either another diagonal matrix or the zero matrix...so, in this way, A is a subspace of Mnn.

Am I correct?
 

Answers and Replies

  • #2
lanedance
Homework Helper
3,304
2
i would just test the subspace requirements directly:

clearly the zero matrix is n the set
now say A,B satisfy AB = BA then is cA+dB in the set? that will satisfy closure under scalar multiplication and addition
 

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